Except, of course, for the differences in length of proof — presumably the chance that a long proof is wrong is less than the chance that a longer proof is wrong, other things being equal.

]]>Anyway, talking about the notion of “almost always true” in mathematics, there is an interesting result in computer science that says that the problem of deciding whether a statement in first-order logic is almost always true can be decided in polynomial space (contrast this to Turing’s result that deciding whether a first-order statement is true is incomputable). Interestingly, if you go to second-order logic, the problem quickly becomes undecidable ðŸ˜‰

]]>These areas of mathematics have many theorems of the form:

“Statement X is true almost everywhere” (ie, true except possibly on a set of measure zero)

OR

“Statement X(n) is eventually true for all n” (ie, there is some k, such that X(n) is true, for all n > k)

OR

“Statement X(n) is true with probability p(n)”

etc.

The greatest achievement of 20th century mathematical statistics — Neyman & Pearson’s theory of statistical inference, which tamed the uncertainty inherent in inductive inference — is a result of this kind. Contemporary experimental and quantitative social science could barely exist without this result, and its descendants.

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